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Derived length of a Frobenius-like kernel

Ercan, Gülin
Güloğlu, İsmail Şuayip
Khukhro, Evgeny
A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide vertical bar FH vertical bar. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C-V(H) of the fixed-point subspace of H by g(m) = 3 + [log(2)(m + 1)]. It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g(r), where r is the sectional rank of C-G(H). As an application, for a finite solvable group G admitting an automorphism phi of prime order coprime to vertical bar G vertical bar, a bound for the p-length of G is obtained in terms of the rank of a Hall p'-subgroup of C(phi). Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its. fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic). (C) 2014 Published by Elsevier Inc.